Abstract

Diffractive optical elements (DOEs) realized by spatial light modulators (SLMs) often have features that distinguish them from most conventional, static DOEs: strong coupling between phase and amplitude modulation, a modulation versus steering parameter characteristic that may not be precisely known (and may vary with, e.g., temperature), and deadspace effects and interpixel cross talk. For an optimal function of the DOE, e.g. as a multiple-beam splitter, the DOE design must account for these artifacts. We present an iterative design method in which the optimal setting of each SLM pixel is carefully chosen by considering the SLM artifacts and the design targets. For instance, the deadspace–interpixel effects are modeled by dividing the pixel to be optimized, and its nearest neighbors, into a number of subareas, each with its unique response and far-field contribution. Besides the customary intensity control, the design targets can also include phase control of the optical field in one or more of the beams in the beam splitter. We show how this can be used to cancel a strong unwanted zeroth-order beam, which results from using a slightly incorrect modulation characteristic for the SLM, by purposely sending a beam in the same direction but with the opposite phase. All the designs have been implemented on the 256 × 256 central pixels of a reflective liquid crystal on silicon SLM with a selected input polarization state and a direction of transmission axis of the output polarizer such that for the available different pixel settings a phase modulation of 2π   rad could be obtained, accompanied by an intensity modulation depth as high as >95%.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Sampsell, "Digital micromirror device and its application to projection displays," J. Vac. Sci. Technol. B 12, 3242-3246 (1994).
    [CrossRef]
  2. A. Gehner, W. Doleschal, A. Elgner, R. Kauert, D. Kunze, and M. Wildenhain, "Active-matrix-addressed micromirror array for wavefront correction in adaptive optics," in MOEMS and Miniaturized Systems II, M. E. Motamedi and R. Goering, eds., Proc. SPIE 4561, 265-275 (2001).
    [CrossRef]
  3. J. L. Pezzaniti and R. A. Chipman, "Phase-only modulation of a twisted nematic liquid-crystal TV by use of the eigenpolarization states," Opt. Lett. 18, 1567-1569 (1993).
    [CrossRef] [PubMed]
  4. B. Löfving, "Self-adjusting dynamic binary phase holograms," Appl. Opt. 36, 2347-2352 (1997).
    [CrossRef] [PubMed]
  5. Z. Zhang, G. Lu, and F. Yu, "Simple method for measuring phase modulation in liquid crystal televisions," Opt. Eng. 33, 3018-3022 (1994).
    [CrossRef]
  6. D. Engström, G. Milewski, J. Bengtsson, and S. Galt, "Diffraction-based determination of the phase modulation for general spatial light modulators," Appl. Opt. 45, 7195-7204 (2006).
    [CrossRef] [PubMed]
  7. D. Prongue, H. Herzig, R. Dandliker, and M. Gale, "Optimized kinoform structures for highly efficient fan-out elements," Appl. Opt. 31, 5706-5711 (1992).
    [CrossRef] [PubMed]
  8. M. W. Farn, "New iterative algorithm for the design of phase-only gratings," in Computer and Optically Generated Holographic Optics; 4th in a Series, I. Cindrich and S. H. Lee, eds., Proc. SPIE 1555, 34-42 (1991).
    [CrossRef]
  9. G.-Z. Yang, B.-Z. Dong, B.-Y. Gu, J.-Y. Zhuang, and O. Ersoy, "Gerchberg-Saxton and Yang-Gu algorithms for phase retrieval in a nonunitary transform system: a comparison," Appl. Opt. 33, 209-218 (1994).
    [CrossRef] [PubMed]
  10. J. Bengtsson, "Design of fan-out kinoforms in the entire scalar diffraction regime with an optimal-rotation-angle method," Appl. Opt. 36, 8435-8444 (1997).
    [CrossRef]
  11. T. Chang, "Proximity effect in electron-beam lithography," J. Vac. Sci. Technol. 12, 1271-1275 (1976).
    [CrossRef]
  12. G. Owen, "Methods for proximity effect correction in electron lithography," J. Vac. Sci. Technol. B 8, 1889-1892 (1990).
    [CrossRef]
  13. F. Nikolajeff, J. Bengtsson, M. Larsson, M. Ekberg, and S. Hard, "Measuring and modeling the proximity effect in direct-write electron-beam lithography kinoforms," Appl. Opt. 34, 897-903 (1995).
    [CrossRef] [PubMed]
  14. A. M. C. Iemmi, I. Moreno, J. Campos, and M. Yzuel, "Anamorphic and spatial frequency dependent phase modulation on liquid crystal displays: optimization of the modulation diffraction efficiency," Opt. Express 13, 2111-2119 (2005).
    [CrossRef] [PubMed]
  15. I. Moreno, J. Campos, C. Gorecki, and M. Yzuel, "Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition," Appl. Opt. 34, 6423-6432 (1995).
  16. M. A. Seldowitz, J. P. Allebach, and D. W. Sweeney, "Synthesis of digital holograms by direct binary search," Appl. Opt. 26, 2788-2798 (1987).
    [CrossRef] [PubMed]
  17. C. Stolz, L. Bigué, and P. Ambs, "Implementation of high-resolution diffractive optical elements on coupled phase and amplitude spatial light modulators," Appl. Opt. 40, 6415-6424 (2001).
    [CrossRef]
  18. N. N. Yoshikawa and T. Yatagai, "Phase optimization of a kinoform by simulated annealing," Appl. Opt. 33, 863-868 (1994).
    [CrossRef] [PubMed]
  19. N. Yoshikawa, M. Itoh, and T. Yatagai, "Quantized phase optimization of two-dimensional Fourier kinoforms by a genetic algorithm," Opt. Lett. 20, 752-754 (1995).
    [CrossRef] [PubMed]
  20. M. Gruneisen, L. DeSandre, J. Rotge, R. Dymale, and D. Lubin, "Programmable diffractive optics for wide-dynamic-range wavefront control using liquid-crystal spatial light modulators," Opt. Eng. 43, 1387-1393 (2004).
    [CrossRef]
  21. J. E. Stockley, D. Subacius, and S. A. Serati, "Influence of the interpixel region in liquid crystal diffraction gratings," in Liquid Crystal Materials, Devices, and Applications VII, R. Shashidhar, ed., Proc. SPIE 3635, 127-136 (1999).
    [CrossRef]
  22. X. Xun and R. W. Cohn, "Phase calibration of spatially nonuniform spatial light modulators," Appl. Opt. 43, 6400-6406 (2004).
    [CrossRef] [PubMed]
  23. J. Bengtsson, "Direct inclusion of the proximity effect in the calculation of kinoforms," Appl. Opt. 33, 4993-4996 (1994).
    [CrossRef] [PubMed]

2006 (1)

2005 (1)

2004 (2)

M. Gruneisen, L. DeSandre, J. Rotge, R. Dymale, and D. Lubin, "Programmable diffractive optics for wide-dynamic-range wavefront control using liquid-crystal spatial light modulators," Opt. Eng. 43, 1387-1393 (2004).
[CrossRef]

X. Xun and R. W. Cohn, "Phase calibration of spatially nonuniform spatial light modulators," Appl. Opt. 43, 6400-6406 (2004).
[CrossRef] [PubMed]

2001 (2)

C. Stolz, L. Bigué, and P. Ambs, "Implementation of high-resolution diffractive optical elements on coupled phase and amplitude spatial light modulators," Appl. Opt. 40, 6415-6424 (2001).
[CrossRef]

A. Gehner, W. Doleschal, A. Elgner, R. Kauert, D. Kunze, and M. Wildenhain, "Active-matrix-addressed micromirror array for wavefront correction in adaptive optics," in MOEMS and Miniaturized Systems II, M. E. Motamedi and R. Goering, eds., Proc. SPIE 4561, 265-275 (2001).
[CrossRef]

1999 (1)

J. E. Stockley, D. Subacius, and S. A. Serati, "Influence of the interpixel region in liquid crystal diffraction gratings," in Liquid Crystal Materials, Devices, and Applications VII, R. Shashidhar, ed., Proc. SPIE 3635, 127-136 (1999).
[CrossRef]

1997 (2)

1995 (3)

1994 (5)

1993 (1)

1992 (1)

1991 (1)

M. W. Farn, "New iterative algorithm for the design of phase-only gratings," in Computer and Optically Generated Holographic Optics; 4th in a Series, I. Cindrich and S. H. Lee, eds., Proc. SPIE 1555, 34-42 (1991).
[CrossRef]

1990 (1)

G. Owen, "Methods for proximity effect correction in electron lithography," J. Vac. Sci. Technol. B 8, 1889-1892 (1990).
[CrossRef]

1987 (1)

1976 (1)

T. Chang, "Proximity effect in electron-beam lithography," J. Vac. Sci. Technol. 12, 1271-1275 (1976).
[CrossRef]

Allebach, J. P.

Ambs, P.

Bengtsson, J.

Bigué, L.

Campos, J.

A. M. C. Iemmi, I. Moreno, J. Campos, and M. Yzuel, "Anamorphic and spatial frequency dependent phase modulation on liquid crystal displays: optimization of the modulation diffraction efficiency," Opt. Express 13, 2111-2119 (2005).
[CrossRef] [PubMed]

I. Moreno, J. Campos, C. Gorecki, and M. Yzuel, "Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition," Appl. Opt. 34, 6423-6432 (1995).

Chang, T.

T. Chang, "Proximity effect in electron-beam lithography," J. Vac. Sci. Technol. 12, 1271-1275 (1976).
[CrossRef]

Chipman, R. A.

Cohn, R. W.

Dandliker, R.

DeSandre, L.

M. Gruneisen, L. DeSandre, J. Rotge, R. Dymale, and D. Lubin, "Programmable diffractive optics for wide-dynamic-range wavefront control using liquid-crystal spatial light modulators," Opt. Eng. 43, 1387-1393 (2004).
[CrossRef]

Doleschal, W.

A. Gehner, W. Doleschal, A. Elgner, R. Kauert, D. Kunze, and M. Wildenhain, "Active-matrix-addressed micromirror array for wavefront correction in adaptive optics," in MOEMS and Miniaturized Systems II, M. E. Motamedi and R. Goering, eds., Proc. SPIE 4561, 265-275 (2001).
[CrossRef]

Dong, B.-Z.

Dymale, R.

M. Gruneisen, L. DeSandre, J. Rotge, R. Dymale, and D. Lubin, "Programmable diffractive optics for wide-dynamic-range wavefront control using liquid-crystal spatial light modulators," Opt. Eng. 43, 1387-1393 (2004).
[CrossRef]

Ekberg, M.

Elgner, A.

A. Gehner, W. Doleschal, A. Elgner, R. Kauert, D. Kunze, and M. Wildenhain, "Active-matrix-addressed micromirror array for wavefront correction in adaptive optics," in MOEMS and Miniaturized Systems II, M. E. Motamedi and R. Goering, eds., Proc. SPIE 4561, 265-275 (2001).
[CrossRef]

Engström, D.

Ersoy, O.

Farn, M. W.

M. W. Farn, "New iterative algorithm for the design of phase-only gratings," in Computer and Optically Generated Holographic Optics; 4th in a Series, I. Cindrich and S. H. Lee, eds., Proc. SPIE 1555, 34-42 (1991).
[CrossRef]

Gale, M.

Galt, S.

Gehner, A.

A. Gehner, W. Doleschal, A. Elgner, R. Kauert, D. Kunze, and M. Wildenhain, "Active-matrix-addressed micromirror array for wavefront correction in adaptive optics," in MOEMS and Miniaturized Systems II, M. E. Motamedi and R. Goering, eds., Proc. SPIE 4561, 265-275 (2001).
[CrossRef]

Gorecki, C.

I. Moreno, J. Campos, C. Gorecki, and M. Yzuel, "Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition," Appl. Opt. 34, 6423-6432 (1995).

Gruneisen, M.

M. Gruneisen, L. DeSandre, J. Rotge, R. Dymale, and D. Lubin, "Programmable diffractive optics for wide-dynamic-range wavefront control using liquid-crystal spatial light modulators," Opt. Eng. 43, 1387-1393 (2004).
[CrossRef]

Gu, B.-Y.

Hard, S.

Herzig, H.

Iemmi, A. M. C.

Itoh, M.

Kauert, R.

A. Gehner, W. Doleschal, A. Elgner, R. Kauert, D. Kunze, and M. Wildenhain, "Active-matrix-addressed micromirror array for wavefront correction in adaptive optics," in MOEMS and Miniaturized Systems II, M. E. Motamedi and R. Goering, eds., Proc. SPIE 4561, 265-275 (2001).
[CrossRef]

Kunze, D.

A. Gehner, W. Doleschal, A. Elgner, R. Kauert, D. Kunze, and M. Wildenhain, "Active-matrix-addressed micromirror array for wavefront correction in adaptive optics," in MOEMS and Miniaturized Systems II, M. E. Motamedi and R. Goering, eds., Proc. SPIE 4561, 265-275 (2001).
[CrossRef]

Larsson, M.

Löfving, B.

Lu, G.

Z. Zhang, G. Lu, and F. Yu, "Simple method for measuring phase modulation in liquid crystal televisions," Opt. Eng. 33, 3018-3022 (1994).
[CrossRef]

Lubin, D.

M. Gruneisen, L. DeSandre, J. Rotge, R. Dymale, and D. Lubin, "Programmable diffractive optics for wide-dynamic-range wavefront control using liquid-crystal spatial light modulators," Opt. Eng. 43, 1387-1393 (2004).
[CrossRef]

Milewski, G.

Moreno, I.

A. M. C. Iemmi, I. Moreno, J. Campos, and M. Yzuel, "Anamorphic and spatial frequency dependent phase modulation on liquid crystal displays: optimization of the modulation diffraction efficiency," Opt. Express 13, 2111-2119 (2005).
[CrossRef] [PubMed]

I. Moreno, J. Campos, C. Gorecki, and M. Yzuel, "Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition," Appl. Opt. 34, 6423-6432 (1995).

Nikolajeff, F.

Owen, G.

G. Owen, "Methods for proximity effect correction in electron lithography," J. Vac. Sci. Technol. B 8, 1889-1892 (1990).
[CrossRef]

Pezzaniti, J. L.

Prongue, D.

Rotge, J.

M. Gruneisen, L. DeSandre, J. Rotge, R. Dymale, and D. Lubin, "Programmable diffractive optics for wide-dynamic-range wavefront control using liquid-crystal spatial light modulators," Opt. Eng. 43, 1387-1393 (2004).
[CrossRef]

Sampsell, J.

J. Sampsell, "Digital micromirror device and its application to projection displays," J. Vac. Sci. Technol. B 12, 3242-3246 (1994).
[CrossRef]

Seldowitz, M. A.

Serati, S. A.

J. E. Stockley, D. Subacius, and S. A. Serati, "Influence of the interpixel region in liquid crystal diffraction gratings," in Liquid Crystal Materials, Devices, and Applications VII, R. Shashidhar, ed., Proc. SPIE 3635, 127-136 (1999).
[CrossRef]

Stockley, J. E.

J. E. Stockley, D. Subacius, and S. A. Serati, "Influence of the interpixel region in liquid crystal diffraction gratings," in Liquid Crystal Materials, Devices, and Applications VII, R. Shashidhar, ed., Proc. SPIE 3635, 127-136 (1999).
[CrossRef]

Stolz, C.

Subacius, D.

J. E. Stockley, D. Subacius, and S. A. Serati, "Influence of the interpixel region in liquid crystal diffraction gratings," in Liquid Crystal Materials, Devices, and Applications VII, R. Shashidhar, ed., Proc. SPIE 3635, 127-136 (1999).
[CrossRef]

Sweeney, D. W.

Wildenhain, M.

A. Gehner, W. Doleschal, A. Elgner, R. Kauert, D. Kunze, and M. Wildenhain, "Active-matrix-addressed micromirror array for wavefront correction in adaptive optics," in MOEMS and Miniaturized Systems II, M. E. Motamedi and R. Goering, eds., Proc. SPIE 4561, 265-275 (2001).
[CrossRef]

Xun, X.

Yang, G.-Z.

Yatagai, T.

Yoshikawa, N.

Yoshikawa, N. N.

Yu, F.

Z. Zhang, G. Lu, and F. Yu, "Simple method for measuring phase modulation in liquid crystal televisions," Opt. Eng. 33, 3018-3022 (1994).
[CrossRef]

Yzuel, M.

A. M. C. Iemmi, I. Moreno, J. Campos, and M. Yzuel, "Anamorphic and spatial frequency dependent phase modulation on liquid crystal displays: optimization of the modulation diffraction efficiency," Opt. Express 13, 2111-2119 (2005).
[CrossRef] [PubMed]

I. Moreno, J. Campos, C. Gorecki, and M. Yzuel, "Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition," Appl. Opt. 34, 6423-6432 (1995).

Zhang, Z.

Z. Zhang, G. Lu, and F. Yu, "Simple method for measuring phase modulation in liquid crystal televisions," Opt. Eng. 33, 3018-3022 (1994).
[CrossRef]

Zhuang, J.-Y.

Appl. Opt. (12)

B. Löfving, "Self-adjusting dynamic binary phase holograms," Appl. Opt. 36, 2347-2352 (1997).
[CrossRef] [PubMed]

G.-Z. Yang, B.-Z. Dong, B.-Y. Gu, J.-Y. Zhuang, and O. Ersoy, "Gerchberg-Saxton and Yang-Gu algorithms for phase retrieval in a nonunitary transform system: a comparison," Appl. Opt. 33, 209-218 (1994).
[CrossRef] [PubMed]

J. Bengtsson, "Design of fan-out kinoforms in the entire scalar diffraction regime with an optimal-rotation-angle method," Appl. Opt. 36, 8435-8444 (1997).
[CrossRef]

D. Engström, G. Milewski, J. Bengtsson, and S. Galt, "Diffraction-based determination of the phase modulation for general spatial light modulators," Appl. Opt. 45, 7195-7204 (2006).
[CrossRef] [PubMed]

D. Prongue, H. Herzig, R. Dandliker, and M. Gale, "Optimized kinoform structures for highly efficient fan-out elements," Appl. Opt. 31, 5706-5711 (1992).
[CrossRef] [PubMed]

F. Nikolajeff, J. Bengtsson, M. Larsson, M. Ekberg, and S. Hard, "Measuring and modeling the proximity effect in direct-write electron-beam lithography kinoforms," Appl. Opt. 34, 897-903 (1995).
[CrossRef] [PubMed]

I. Moreno, J. Campos, C. Gorecki, and M. Yzuel, "Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition," Appl. Opt. 34, 6423-6432 (1995).

M. A. Seldowitz, J. P. Allebach, and D. W. Sweeney, "Synthesis of digital holograms by direct binary search," Appl. Opt. 26, 2788-2798 (1987).
[CrossRef] [PubMed]

C. Stolz, L. Bigué, and P. Ambs, "Implementation of high-resolution diffractive optical elements on coupled phase and amplitude spatial light modulators," Appl. Opt. 40, 6415-6424 (2001).
[CrossRef]

N. N. Yoshikawa and T. Yatagai, "Phase optimization of a kinoform by simulated annealing," Appl. Opt. 33, 863-868 (1994).
[CrossRef] [PubMed]

X. Xun and R. W. Cohn, "Phase calibration of spatially nonuniform spatial light modulators," Appl. Opt. 43, 6400-6406 (2004).
[CrossRef] [PubMed]

J. Bengtsson, "Direct inclusion of the proximity effect in the calculation of kinoforms," Appl. Opt. 33, 4993-4996 (1994).
[CrossRef] [PubMed]

J. Vac. Sci. Technol. (1)

T. Chang, "Proximity effect in electron-beam lithography," J. Vac. Sci. Technol. 12, 1271-1275 (1976).
[CrossRef]

J. Vac. Sci. Technol. B (2)

G. Owen, "Methods for proximity effect correction in electron lithography," J. Vac. Sci. Technol. B 8, 1889-1892 (1990).
[CrossRef]

J. Sampsell, "Digital micromirror device and its application to projection displays," J. Vac. Sci. Technol. B 12, 3242-3246 (1994).
[CrossRef]

Opt. Eng. (2)

Z. Zhang, G. Lu, and F. Yu, "Simple method for measuring phase modulation in liquid crystal televisions," Opt. Eng. 33, 3018-3022 (1994).
[CrossRef]

M. Gruneisen, L. DeSandre, J. Rotge, R. Dymale, and D. Lubin, "Programmable diffractive optics for wide-dynamic-range wavefront control using liquid-crystal spatial light modulators," Opt. Eng. 43, 1387-1393 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Proc. SPIE (3)

J. E. Stockley, D. Subacius, and S. A. Serati, "Influence of the interpixel region in liquid crystal diffraction gratings," in Liquid Crystal Materials, Devices, and Applications VII, R. Shashidhar, ed., Proc. SPIE 3635, 127-136 (1999).
[CrossRef]

M. W. Farn, "New iterative algorithm for the design of phase-only gratings," in Computer and Optically Generated Holographic Optics; 4th in a Series, I. Cindrich and S. H. Lee, eds., Proc. SPIE 1555, 34-42 (1991).
[CrossRef]

A. Gehner, W. Doleschal, A. Elgner, R. Kauert, D. Kunze, and M. Wildenhain, "Active-matrix-addressed micromirror array for wavefront correction in adaptive optics," in MOEMS and Miniaturized Systems II, M. E. Motamedi and R. Goering, eds., Proc. SPIE 4561, 265-275 (2001).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Coupled amplitude and phase modulation characteristics as functions of pixel setting L (taking on integer values in the 0–255 range), as measured for our SLM device at the 543   m wavelength that was used in all the designs and experiments.

Fig. 2
Fig. 2

Flow chart outlining the design algorithm.

Fig. 3
Fig. 3

(Color online) Measured far-field intensity distributions for DOEs designed using (a) A ( L ) 1 , (b) A(L) from measurements, (c) A(L) from measurements and additionally using active zeroth-order suppression.

Fig. 4
Fig. 4

(a) Evolution of the simulated intensity with the number of iterations in the 4 × 4 positions and zeroth order during the DOE design. (b) Evolution of the simulated phase of the zeroth-order field, with a target value of 45°. Inset, a portion of the SLM from the final iteration, with the gray scale representing pixel setting L.

Fig. 5
Fig. 5

(Color online) Solid curve shows the measured relative zeroth-order intensity I 0 / I m ¯ for 16 DOEs designed using the same nonzero target intensity I 0 ^ target but a different phase of the field ϕ 0 ^ target in the zeroth order. All the DOEs have the same target intensities I m target for the 4 × 4 array. As a comparison, the dashed line shows I 0 / I m ¯ for a DOE designed with zero target intensity in the zeroth order. In the insets are three examples of the measured intensity distribution in the far field. In the design of all 17 DOEs, the same, but slightly incorrect, A(L) and ϕ ( L ) relations were used so that a strong uncontrolled zeroth-order intensity was obtained despite active zeroth-order suppression ( I 0 target = 0 ) in the design of the DOE corresponding to the dashed line.

Fig. 6
Fig. 6

Subarea pixel model for one whole pixel (enclosed by the inner dashed line) and for parts of its neighboring pixels. The outer dashed line encloses all the subareas whose effective settings change when changing the setting of the pixel enclosed by the inner line. The number in the center of each subarea is the value L s k eff of the effective setting. For the three types of dependent subarea (side, corner, and deadspace), the boxes show the optimized weights used to obtain their L s k eff values, and the percentages indicate how much the setting of a neighboring pixel or deadspace area influences the L s k eff value. The deadspace box also shows L ds , the intrinsic setting of the deadspace. The subarea width, which is also a parameter in the model optimization, is also indicated.

Fig. 7
Fig. 7

(Color online) Measured and simulated values of the relative zeroth-order intensity I 0 / I m ¯ for 21 different test DOEs. Simulation A is the parameter set that agrees best with the measurements. In simulations B and C, an identical set of parameters was used except that the subarea width was changed by 1.0 μ m and + 1.0 μ m , respectively, from the value in simulation A.

Fig. 8
Fig. 8

(Color online) (a) Measured far-field for a DOE designed using the full subarea pixel model with the same design task of 4 × 4 spots and a suppressed zeroth order as previously. (b) Same as (a) but the DOE is designed to produce a large-angle ring. (c) Same captured image as in (b) but with a tenfold intensity saturation to show the background intensity distribution. The horizontal lines are caused by electrical disturbances in the camera.

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

E k m ( L k ) = E k inc U k m A ( L k ) exp { j ϕ ( L k ) } ,
U k m = 1 4 π ( j | k | D m r k m { j | k | 1 r k m } ) 4 r k m exp ( j | k | r k m ) × sin ( k ˜ x ( a / 2 ) ) sin ( k ˜ y ( b / 2 ) ) k ˜ x k ˜ y ,
k ˜ x = k x , k inc + | k | ( x k u m ) r k m ,
k ˜ y = k y , k inc + | k | ( y k v m ) r k m ,
E m = k ^ k E k ^ m ( L k ^ ) + E k m ( L k ) = k ^ E k ^ m ( L k ^ ) + [ E k m ( L k ) E k m ( L k ) ] = E m + E k inc U k m [ A ( L k ) exp { j ϕ ( L k ) } A ( L k ) exp { j ϕ ( L k ) } ] ,
Q = m [ I m I m target ] 2 ,
ϵ uni = max m [ I m / I m target ] min m [ I m / I m target ] max m [ I m / I m target ] + min m [ I m / I m target ]
E k m = p = 1 M q = 1 M E k m sub - DOE ( p , q ) ,
Q = m [ I m I m target ] 2 + m ^ | A m ^ exp { j ϕ m ^ } A m ^ target × exp { j ϕ m ^ target } | 4 .
E m = k E k inc [ s U s k m A ( L s k eff ) exp { j ϕ ( L s k eff ) } ] ,
E m = E m + s = s ^ [ E k inc U s k m [ A ( L s k e f f ) exp { j ϕ ( L s k e f f ) } A ( L s k eff ) exp { j ϕ ( L s k eff ) } ] ] ,

Metrics